Distributional Invariances and Interventional Markov Equivalence for Mixed Graph Models

The invariance properties of interventional distributions relative to the observational distribution, and how these properties allow us to refine Markov equivalence classes (MECs) of DAGs, is central to causal DAG discovery algorithms that use both interventional and observational data. Here, we show how the invariance properties of interventional DAG models, and the corresponding refinement of MECs into interventional MECs, can be generalized to mixed graphical models that allow for latent cofounders and selection variables. We first generalize interventional Markov equivalence to all formal independence models associated to loopless mixed graphs. For ancestral graphs, we prove the resulting interventional MECs admit a graphical characterization generalizing that of DAGs. We then define interventional distributions for acyclic directed mixed graph models, and prove that this generalization aligns with the graphical generalization of interventional Markov equivalence given for the formal independence models. This provides a framework for causal model discovery via observational and interventional data in the presence of latent confounders that applies even when the interventions are uncontrolled.